Methods, systems, and computer-readable storage media for enhanced phase-contrast x-ray imaging

ABSTRACT

Systems and methods that directly image attenuation-based object grid, use a source grid to improve imaging of the object grid using a high-energy polychromatic source, and use a detector grid having gratings oriented substantially orthogonally to that of the object grid, can address artifacts and beam hardening effects that limit the quality and discriminatory power of high-energy x-ray imaging that includes phase contrast.

CROSS REFERENCE TO RELATED APPLICATIONS

This is the U.S. National Stage of International Application No.PCT/US2020/023884, filed Mar. 20, 2020, which was published in Englishunder PCT Article 21(2), which is a continuation of U.S. patentapplication Ser. No. 16/363,989, filed Mar. 25, 2019, now issued as U.S.Pat. No. 11,006,912. The non-provisional application is incorporatedherein in its entirety.

ACKNOWLEDGEMENT OF GOVERNMENT SUPPORT

This invention was made with Government support under ContractDE-AC0576RL01830 awarded by the U.S. Department of Energy. TheGovernment has certain rights in the invention.

FIELD

The present disclosure relates to phase-contrast x-ray imaging, and moreparticularly to phase-contrast x-ray imaging having improved quality andmaterial discriminating power.

BACKGROUND

X-ray based imaging is used in a variety of non-destructive examination(NDE) applications. In many of these applications, which can range frommedical imaging to security screening, the primary x-ray characteristicsare density and effective atomic number derived from multispectral x-rayattenuation measurements. The addition of phase contrast as a thirdimaging characteristic can improve material discrimination by detectionof refractive and scattering effects in examined objects. However,unwanted spectral effects, misleading image artifacts, and the demandsassociated with producing images with increased material penetrationrequired novel solutions in order to improve resultant x-ray images.Such solutions are described herein.

SUMMARY

Disclosed are methods, systems, and non-transitory, computer-readablestorage media storing programs for phase-contrast x-ray imaging havingimproved quality and material discriminating power.

In some embodiments, a method comprises emitting source x-rays from apolychromatic source operating at an endpoint energy greater than orequal to 100 keV and generating a spot size greater than or equal to 0.5mm; creating a series of periodically repeating apparent sources fromthe source x-rays using a source grating; patterning the series ofperiodically repeating apparent sources into a patterned beam using anobject grating placed proximal to an object to be imaged and atdistances L₁ from the source grating and L₂ from a detector grating,wherein the periodicities, P, of the source and object grating elementsare related by P_(source)=P_(object)*[(L₁+L₂)/L₂] and wherein the sourceand object grating elements are substantially parallel; acquiringthrough the detector grating a first image with the object and a secondimage without the object, wherein the detector grating is orientedsubstantially orthogonally relative to the object grating and beam axisand wherein the object grating and the detector grating have asubstantially equivalent x-ray attenuating factor; measuringvisibilities of the object grating from the first and second images todetermine an object grating visibility reduction due to scatter and beamhardening; measuring visibilities of the detector grating from the firstand second images to determine a detector grating visibility reductiondue to beam hardening; and applying a beam hardening correction based ona comparison of the object grating visibility reduction and the detectorgrating visibility reduction to generate a corrected scatter image.

In certain embodiments, the method can further comprise operating thepolychromatic source at an endpoint energy greater than or equal to 150keV, 160 keV, 175 keV, 200 keV, or 450 keV. In certain embodiments, themethod can further comprise tilting the object grating and detectorgrating by rotating the gratings about an axis parallel to gratingelement lines. In certain embodiments, the method can further comprisetilting the source grating by rotating the gratings about an axisparallel to grating element lines.

In certain embodiments, the object grating is approximately equidistantbetween the source and the detector. In certain embodiments, thedetector grating has a periodicity. P_(detector), equivalent to that ofthe source grating, P_(source). In certain embodiments, the object anddetector gratings comprise an equivalent material and have an equivalentthickness. In certain embodiments, the source grating, object grating,detector grating, or combinations thereof have grating elementscomprising a parallel line pattern.

In certain embodiments, the object to be imaged is a scatter test objectcalibration standard and further comprising performing a calibration ofx-ray scatter, the scatter test object calibration standard comprisingmetal or metal oxide particles distributed in a polymer matrix andhaving a stepped-wedge geometry of at least three different thicknesses.In certain embodiments, the object to be imaged is a beam hardening testobject calibration standard and further comprising performing acalibration of beam hardening, the beam hardening test objectcalibration standard comprising three or more homogeneous materials in arange of atomic numbers, with no large density variations on lengthscales between 10 nm and 200 microns, and have a thickness such that10-90% of the x-ray intensity is transmitted through the test object.

In some embodiments, a system comprises a polychromatic x-ray sourceconfigured to provide source x-rays at an endpoint energy greater thanor equal to 100 keV and a spot size greater than 0.5 mm; a sourcegrating configured to create a series of periodically repeating apparentsources from the source x-ray; an object grating proximal to a positionof an object to be imaged and at distances L₁ from the source gratingand L₂ from a detector grating, wherein the periodicities of the sourceand object gratings are related by P_(source)=P_(object)*[(L₁+L₂)/L₂],the object grating configured to pattern the series of periodicallyrepeating apparent sources into a patterned beam; and a detector gratinghaving detector grating elements that are oriented orthogonally relativeto object grating elements and a beam axis, the detector and objectgratings having an equivalent x-ray attenuation factor. The systemfurther comprises processing circuitry operably connected to thedetector and configured to execute computer-readable instructions toacquire through the detector grating a first image with the object and asecond image without the object; measure visibilities of the objectgrating from the first and second images to determine an object gratingvisibility reduction due to scatter and beam hardening; measurevisibilities of the detector grating from the first and second images todetermine a detector grating visibility reduction due to beam hardening;and apply a beam hardening correction based on a comparison of theobject grating visibility reduction and the detector grating visibilityreduction to generate a corrected scatter image.

In certain embodiments, the polychromatic source is configured toprovide source x-rays at an endpoint energy greater than or equal to 150keV, 160 keV, 175 keV, 200 keV, or 450 keV. In certain embodiments, theobject grating and detector grating are positioned such that objectgrating elements and detector grating elements are tilted by a rotationof the gratings about an axis parallel to grating element lines. Incertain embodiments, the source grating is positioned such that sourcegrating elements are tilted by a rotation of the gratings about an axisparallel to grating element lines. In certain embodiments, the objectgrating is positioned approximately equidistant between the source andthe detector. In certain embodiments, the detector grating has aperiodicity, P_(detector), equivalent to that of the source grating,P_(source). In certain embodiments, the detector grating abuts thedetector. In certain embodiments, the object and detector gratingscomprise an equivalent material and have an equivalent thickness. Incertain embodiments, the source grating, object grating, detectorgrating, or combinations thereof have grating elements comprising aparallel line pattern.

In some embodiments, a non-transitory computer readable storage mediumstores one or more programs, the one or more programs compriseinstructions, which when executed by one or more processors operablyconnected to an x-ray imaging system, cause the system to acquirethrough the detector grating a first image with the object and a secondimage without the object; measure visibilities of the object gratingfrom the first and second images to determine an object gratingvisibility reduction due to scatter and beam hardening; measurevisibilities of the detector grating from the first and second images todetermine a detector grating visibility reduction due to beam hardening;and apply a beam hardening correction based on a comparison of theobject grating visibility reduction and the detector grating visibilityreduction to generate a corrected scatter image. The x-ray imagingsystem to which the processor(s) are operably connected comprise apolychromatic x-ray source configured to provide source x-rays at anendpoint energy greater than or equal to 100 keV and a spot size greaterthan 0.5 mm, a source grating configured to create a series ofperiodically repeating apparent sources from the source x-ray; an objectgrating proximal to a position of an object to be imaged band atdistances L₁ from the source grating and L₂ from a detector grating,wherein the periodicities of the source and object gratings are relatedby P_(source)=P_(object)*[(L₁+L₂)/L₂], the object grating configured topattern the series of periodically repeating apparent sources into apatterned beam; and a detector grating having detector grating elementsthat are oriented orthogonally relative to object grating elements and abeam axis, the detector and object gratings having an equivalent x-rayattenuation factor.

In certain embodiments, the non-transitory computer readable storagemedium storing one or more programs, the one or more programs comprisinginstructions, which when executed by one or more processors operablyconnected to the x-ray imaging system further cause the x-ray imagingsystem to perform a calibration, wherein the object to be imaged is ascatter test object, a beam hardening test object, or both.

The purpose of the foregoing summary and the latter abstract is toenable the United States Patent and Trademark Office and the publicgenerally, especially the scientists, engineers, and practitioners inthe art who are not familiar with patent or legal terms or phraseology,to determine quickly from a cursory inspection the nature and essence ofthe technical disclosure of the application. Neither the summary nor theabstract is intended to define the invention of the application, whichis measured by the claims, nor is it intended to be limiting as to thescope of the claims in any way.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B include a schematic diagram and a flow chart,respectively, depicting embodiments described herein.

FIG. 2 is a diagram of one embodiment of a computational system forenhanced phase-contrast x-ray imaging and/or beam-hardening correction.

FIG. 3 includes photos of one embodiment of an arrangement of source,object, and detector gratings in an enhanced phase-contrast x-rayimaging system.

FIG. 4 is a screenshot of one example of image acquisition andprocessing software.

FIG. 5 includes an image of a Fourier transform of an image obtained byenhanced phase-contrast x-ray imaging as described herein.

FIG. 6 is a graph showing corrected scatter ratio as a function of gridfrequency (in LPI) for two different particle sizes.

FIG. 7 is a graph showing fringe visibility as a function of endpointenergy and grid spatial frequency.

FIG. 8 is a graph showing visibility reduction as a function of endpointenergy, for a 1.26 cm Al sample (shown with and without the beamhardening correction).

FIG. 9 is a graph showing corrected scatter ratio as a function ofparticle size, for a 60 kV spectrum. Theoretical predictions are shownas solid lines; measurements are indicated as points with error bars

FIGS. 10A and 10B include images of a small bag (8″×9″) as in a securityscreening context. FIG. 10A is a conventional attenuation image. FIG.10B is an attenuation image with corrected scatter overlay according toembodiments described herein. Propellants present in the bag showscatter, as do business cards, a watch band, and tic tac candies.Chapstick next to the model rocket engine looks similar by shape, butdoes not exhibit scatter.

FIGS. 11A and 11B are images related to one embodiment of a calibrationstandard. FIG. 11A is a photograph of ZnO scatter step wedge calibrationstandards. FIG. 11B are beam-hardening corrected scatter images taken athigh energy (160 kV with 2 mm Cu and 2 mm Al filtration) and at lowenergy (100 kV with 2 mm Al filtration).

FIG. 12 is a photo of an embodiment of a system described herein (absenta source grid). An X-ray tube is at lower right and the beam directionis from lower right to upper left. Filter materials are immediately infront of the x-ray tube; the standard test objects can be seen halfwaydown the table at upper left; the object and detector grids are alsovisible in the upper left quadrant. The detector is obscured by thedetector grid.

FIG. 13 is a graph showing calculated beam spectra at “high” energy and“low” energy.

FIG. 14 shows a set of vials containing samples set for end-on imagingand a single vial containing explosive material.

FIG. 15 is a graph showing Dual Energy Results. The y-axis indicatesμ_(L)/μ_(H), which can be related to Z_(eff), and the x-axis is μ_(H),which can be related to density. Materials A-D represent four differentexplosive materials and encompass a number of preparations.

FIG. 16 is a graph showing scatter coefficient at low energy ν_(L) as afunction of μ_(H) (proportional to density). Explosives are labeled byletters A-D.

FIG. 17 is a graph showing scatter coefficient at high energy ν_(H) as afunction of μ_(H) (proportional to density). Explosives are labeled byletters A-D.

FIGS. 18A and 18B are (18A) close-up of a region of the grating image,showing the vertical grating lines and the longer period Moire pattern.Near the center is a line of bad pixels from a scratch on the detectorand (18B) close up of a grating image with both the object and detectorgratings, in a section of bad pixels.

FIGS. 19A-19C are regions of a Fourier transform of a grating image.

FIG. 20 is a Fourier transform of a grating image with both an objectand detector grid, showing first and second harmonics as well ascross-harmonics.

FIGS. 21A and 21B are images before and after bad pixel correction of aline of bad pixels, respectively.

FIGS. 22A and 22B are images before and after bad pixel correction of ablob, respectively.

FIGS. 23A and 23B are images before and after bad pixel correction of ascatter image of a region illustrated in FIG. 21A.

FIGS. 24A and 24B are images of a region of bad pixels with both adetector and an object grid, and with contributions from the secondharmonic peaks of both grids before and after bad pixel correction,respectively.

FIG. 25 is a schematic illustrating grid (i.e., grating) tilt. The beamdirection is indicated by the arrow 2502.

FIGS. 26A and 26B are graphs showing modeled flux at the detector from a0.4 mm spot size 100 kVp source showing values of fringe visibility fits(26A), and plot of fringe visibility (H₁/H₀) along with the secondharmonic ratio (H₂/H₀) as a function of rotation angle (26B).

FIGS. 27A and 27B are graphs showing modeled flux at the detector from a0 mm spot size 100 kVp source showing values of fringe visibility fits(27A), and plot of analytical fringe visibility (H₁/H₀) for perfect (Pb)attenuation and a parallel beam source (27B).

FIG. 28 is a graph showing the modeled fits to fringe visibilitymeasurements with a tilted object grating.

DETAILED DESCRIPTION

Phase contrast x-ray imaging or gratings-based phase contrast imagingcan allow for detection of small deviations in the direction of an x-rayas it passes through a material. These deflections, specificallyscatter, can be used to detect texture in a material, such as a powderor a composite, below the imaging resolution of the system. Theinventors have determined that measurements at high energies can providescatter signatures indicative of sub-resolution texture within a samplein order to help identify materials and that the systems, methods, andstorage media described herein can be relevant for applications rangingfrom medical imaging to materials characterization to securityscreening.

Embodiments described herein can be utilized for discernment ofmaterials properties, especially in non-destructive examinationapplications. For example, material wetting or compression could beexamined (e.g., concrete, plaster, materials that are formed throughcompression of powders), as could fiber orientations in materials madefrom carbon fibers or other fibrous materials. Medical applications arealso possible, including diagnostic imaging with either radiography orCT. Scatter has been shown to give enhanced contrast for lung structureand for bones. Finally, additional security screening applications maybe possible, such as detecting 3-D fabricated parts based on texture orlocating powdered materials in mail screening or vehicle screening.Conventional airport security relies on dual-energy x-ray images thatcan be used to estimate material density and effective atomic number;these two features are relied upon to discriminate threat objects frombenign consumer products. However, the estimation from conventionalsecurity scanners is often insufficient to effectively distinguish andidentify threat objects. Phase contrast imaging can provide additionalmaterials signatures from x-ray measurements: attenuation, which issimilar to a conventional x-ray image; refraction or phase, which isbased on electron density variations and can be sensitive to low-Zmaterials; and scatter, which detects the presence of texture (such aspowders or composites) below the imaging resolution of the system. Theaddition of new signatures increases the number of features which can beused for material discrimination, potentially reducing false alarm ratesduring security screening. Furthermore, one mode may have a lowerdetection limit than absorption, enabling the detection/identificationof additional items and/or features.

Current phase contrast imaging systems typically rely on a grid whichproduces an x-ray interference pattern (typically with a period of a fewmicrons) and an analyzer grid matched to the undistorted interferencepattern. These systems require sub-micron stability and are verydifficult to scale to higher, more penetrating energies; they oftenoperate at energies below 100 kVp. When grid fabrication for energiesabove 100 kVp is possible, it is difficult and expensive. First, theperiod should be smaller than the coherence length (which decreases asenergy increases). Second, the thickness of the attenuating parts of thegrid need to be thick enough to stop the x-rays, and this becomes largerat high energies. The net effect is that fabrication with fine featuresizes but extremely large aspect ratios are required; something that isoften impractical to manufacture.

For aviation security, phase contrast imaging is not currently used.Dual energy systems provide estimates of material density and effectiveatomic number to help discriminate benign materials from threats. Addingphase contrast would allow refraction information and textureinformation to be measured in addition to dual energy, providing abroader basis of material signatures for discriminating materials, andpotentially reducing false alarm rates.

Embodiments described herein can detect sub-resolution texture using anobject grid as a patterning object in the beam and at a standoffdistance from the detector, where the image of the object grid isprojected. If a sample containing sub-resolution density variations(such as a powder) is placed near the object grid, the refractive indexvariations within the object will cause deflections of the x-ray beam,ultimately causing blurring of the projected object grid pattern. Thiscan be described as a reduction in visibility of the pattern.

Importantly, traditional phase contrast imaging occurs at relatively lowenergies (<100 kVp). Embodiments described herein measure x-rayrefraction and scatter at higher energies, while making corrections forspectral effects which can cause spurious scatter-like signals. Theembodiments enable the use of high energies which are relevant for NDEapplications including airport screening (e.g. 160 kVp). Indeed, theinventors have measured scatter at energies as high as 450 kVp.

The x-ray energies referred to herein are endpoint energies. An x-raytube produces a polychromatic spectrum of x-rays, with a peak energydefined by the electron energy impinging upon the anode. As the x-rayspass through the object, some energies are more readily absorbed thanothers, which means that the spectrum behind the object is differentthan the original spectrum. This in turn changes the visibility of thegrid lines. For most materials in the range of energies described hereinfor x-ray imaging, higher energies are more penetrating than lowerenergies, which are more readily absorbed in materials. This means thatthe original visibility of an object grid will tend to be higher forlower energies. When a polychromatic beam passes through an object, thelower energies in the spectrum are more readily absorbed, an effectreferred to as “beam hardening”. In this case, the inventors havedetermined that since the resulting spectrum has more intensity at highenergy than the original spectrum, this will cause a reduction in thevisibility of the object grid, even in the absence of actual scatteringin the object. As the system is run at higher energies and used tointerrogate more attenuating objects, the changes in beam spectrumcaused by object attenuation also lead to changes in grid patternvisibility, which must be corrected for in order to isolate thevisibility reduction due to scatter.

Embodiments described herein differ from other three-grid combinationsystems and methods at least because some embodiments enable high-energyoperation using a polychromatic radiography source (e.g., energies above100 kVp, 125 kVp, 150 kVp, 160 kVp, 175 kVp, 200 kVp, or 450 kVp, with aspot size, defined as the spatial extent of the region on the x-ray tubeanode from which x-rays are emitted, of at least 0.5 mm), in contrast toa synchrotron source or a conventional source operated at lower energiesand/or spot sizes. In certain embodiments, the source operates with acurrent ranging from 0.1 to 1000 milliamps.

In particular, embodiments described herein differ from a three-gridTalbot-Lau interferometer, which uses a source grid to increase spatialcoherence, an object grid that forms an interference pattern, and ananalyzer grid that detects small changes in the very small interferencepattern. In other words, the object grid is a phase element and sets upan interference pattern that impinges on the analyzer grid in order tohelp detect deviations in the interference pattern without resolving itdirectly. The source grid in the Talbot-Lau configuration is required toform a sufficiently smooth wave front to establish an interferencepattern and the required coherence. In contrast, embodiments describedherein utilize large spot sizes (at least 0.5 mm) while retaining theability to not blur the pattern image and to improve resolution, notcoherence. The Talbot-Lau analyzer grating is aligned with the objectgrating and matches the projected object grating period. Anotherdistinction of present embodiments compared to a Talbot-Lau-styleinterferometer is the absence of a requirement for gratings which areboth fine (period of 5 microns or less) and extremely high aspect ratio(often 10:1, and up to 100:1 for 100 keV), sub-micron alignment andstability, and highly precise (sub-micron) stepping of an analyzer gridplaced near the detector. This combination is difficult and impracticalfor many applications for conventional systems. In contrast, someembodiments described herein utilize gratings having grating elementscomprising parallel channels with an aspect ratio less than 10:1, 8:1,5:1, or 3:1 when the source operates at an energy of at least 100 kVp.In certain embodiments, the gratings can have a scale greater than a 2micron period, a 5 micron period, a 10 micron period, a 25 micronperiod, a 50 micron period, or a 100 micron period, which can enabledifferent fabrication methods that are much easier.

In summary, the inventors have determined that the combination of adirectly imaged, attenuation-based object grid, the use of a source gridto improve imaging of the object grid using a high-energy polychromaticsource with a large spot size, and the use of a stationary detector gridhaving gratings oriented substantially orthogonally to that of theobject grid, addresses the artifacts and beam hardening effects thatlimit the quality and discriminatory power of high-energy x-ray imagingthat includes phase contrast. The object grid is visible on the detectedimage and is, therefore, sufficiently coarse to be directly visualizedon the detector. However, this coarseness can reduce scattersensitivity. In certain embodiments, the object grid is positionedsubstantially equidistant between the source and detector in order tooptimize contrast for most samples by providing 2× magnification of thegrid on the detector. Furthermore, most high energy sources have a largex-ray tube spot size, so their use is enabled by the added source grid.Finally, high energy applications typically involve highly attenuatingobjects, making the beam hardening correction critical for accurateresults, which requires the detector grid. Thus, all three grids operatesynergistically to enable embodiments disclosed herein.

The explanations of terms and abbreviations herein are provided tobetter describe the present disclosure and to guide those of ordinaryskill in the art in the practice of the present disclosure. As usedherein, “comprising” means “including” and the singular forms “a” or“an” or “the” include plural references unless the context clearlydictates otherwise. The term “or” refers to a single element of statedalternative elements or a combination of two or more elements, unlessthe context clearly indicates otherwise.

Unless explained otherwise, all technical and scientific terms usedherein have the same meaning as commonly understood to one of ordinaryskill in the art to which this disclosure belongs. Although methods andmaterials similar or equivalent to those described herein can be used inthe practice or testing of the present disclosure, suitable methods andmaterials are described below. The materials, methods, and examples areillustrative only and not intended to be limiting. Other features of thedisclosure are apparent from the following detailed description and theclaims.

Unless otherwise indicated, all numbers expressing quantities ofcomponents, molecular weights, distances, energies, percentages,temperatures, times, and so forth, as used in the specification orclaims are to be understood as being modified by the term “about.”Accordingly, unless otherwise implicitly or explicitly indicated, orunless the context is properly understood by a person of ordinary skillin the art to have a more definitive construction, the numericalparameters set forth are approximations that may depend on the desiredproperties sought and/or limits of detection under standard testconditions/methods as known to those of ordinary skill in the art. Whendirectly and explicitly distinguishing embodiments from discussed priorart, the embodiment numbers are not approximations unless the word“about” is recited.

Referring to FIG. 1A, a diagram summarizes one embodiment of a systemfor beam-hardening-corrected, phase-contrast, x-ray imaging. Asillustrated, the embodiment comprises three x-ray anti-scatter gridshaving parallel line grating elements. The source grid 106 is placed atthe x-ray source 100, the object grid 102 is in proximity to an object103 being imaged, and the detector grid 104 abuts the detector 105. Thesource and detector are placed some distance apart, with an object beingimaged located between the source and the detector. The source isoperated at high energies and generates a large spot size. The objectgrid can be on the source or detector side of the object and the objectgrid lines are spaced at period, P_(object). The source grid is placednear the x-ray source with period, P_(source) substantially equal toP_(object)·[(L₁+L₂)/L₁], wherein L₁ and L₂ are distances from the sourceto the object and from the object to the grid, respectively. In certainembodiments, the distance between the source and the object and thedistance between the object and detector are substantially equivalent.In certain embodiments, the detector grating has a periodicity that isequivalent to that of the source grating. The fundamental relationshipis between P_(source) and P_(object), to get the patterns to overlaywhen projected onto the detector. As a matter of convenience,efficiency, and/or optimization to set P_(detector) equal to theprojected P_(object) to make the regions around their respective Fourierpeaks are about the same size in Fourier space. When L₁ and L₂ areequal, P_(source) can equal P_(detector).

The source grid can compensate for a large x-ray tube spot size, whichcan be detrimental to being able to resolve the lines of the objectgrid. In other words, the source grid can allow the object grid to beimaged on the detector more clearly. The detector grid is placed at thedetector and the orientation of the grating elements is substantially 90degrees rotated relative to those of the object grid. With regard torotation of the detector grating elements, substantially can refer to anerror of ±1, ±2, or ±5 degrees. In certain embodiments, the error indetector grating rotation angle should be less than that which wouldavoid overlap of the first harmonics in Fourier space.

The object grid and detector grid have substantially the sameattenuating factor. With regard to the attenuating factor, substantiallyrefers to an error of ±1%, ±3%, ±5%, or +10%. For example, the objectand detector grids can comprise the same material and same thickness.The detector grid is used to correct for artifacts caused by beamhardening, where the spectrum of the beam is changed by attenuatingobjects. When the object grid is substantially equidistant from thesource and detector, the source and detector gratings can havesubstantially the same grating element period.

FIG. 1B includes a flowchart summarizing one example of acomputer-implemented method of beam-hardening correction ofphase-contrast x-ray imaging. The method can be embodied bynon-transitory, computer-readable storage media storing instructionsthat can be executed to perform the method. In the illustrated example,first and second images are acquired with and without the object to beimaged 151. The “first image” and “second image” terms do not refer tochronological sequence; the images can be acquired in any order. Theobject grating visibilities are measured in both images 152. Similarly,the detector grating visibilities are measured in both images 153. Thevisibilities are compared 154 to determine the object grating visibilityreduction and the detector grating visibility reduction. Based on thecomparison of the reductions 154, a beam-hardening correction is applied155 to yield a corrected phase-contrast x-ray image.

Non-transitory as used herein when referring to a computer-readablestorage medium, is a limitation of the medium itself (i.e., tangible,not a propagating electromagnetic signal) as opposed to a limitation ondata storage persistency. The term is not intended to otherwise limitthe type of physical computer-readable storage device that isencompassed by the phrase computer-accessible medium or memory. Forinstance, the terms “non-transitory computer readable medium” or“tangible memory” are intended to encompass types of storage devicesthat do not necessarily store information permanently, including but notlimited to, computer-readable media that store data only for shortperiods of time and/or only in the presence of power, such as registermemory, processor cache and Random Access Memory (RAM). Programinstructions and data stored on a tangible computer-accessible storagemedium in non-transitory form may further be transmitted by transmissionmedia or signals such as electrical, electromagnetic, or digitalsignals, which may be conveyed via a communication medium such as anetwork and/or a wireless link.

FIG. 2 is one embodiment of a computational system or computingenvironment to which an enhanced phase-contrast x-ray imaging system canbe operably connected. Alternatively, the computational system can beintegrated within an enhanced phase-contrast x-ray imaging system. Inone example, a computing environment such as shown in FIG. 2 can be usedto control operation of the imaging system. The computing environmentcan further be used to acquire images and to perform beam-hardeningcorrections as described elsewhere herein.

With reference to FIG. 2 , an example system for implementing someembodiments includes a general-purpose computing device in the form of acomputer 210. Components of computer 210 may include, but are notlimited to, a processing unit 220 (which is not limited to CPUs, but cancomprise GPUs), a system memory 230, and a system bus 221 that couplesvarious system components including the system memory to the processingunit 220. The system bus 221 may be any of several types of busstructures including a memory bus or memory controller, a peripheralbus, and a local bus using any of a variety of bus architectures. By wayof example, and not limitation, such architectures include IndustryStandard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus,Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA)local bus, and Peripheral Component Interconnect (PCI) bus also known asMezzanine bus. Memory and programs described herein be deployed incorresponding portions of FIG. 2 .

Computer 210 typically includes a variety of computer readable media.Computer readable media can be any available media that can be accessedby computer 210 and includes both volatile and nonvolatile media,removable and non-removable media. By way of example, and notlimitation, computer readable media may comprise computer storage mediaand communication media. Computer storage media is different from, anddoes not include, a modulated data signal or carrier wave. It includeshardware storage media including both volatile and nonvolatile,removable and non-removable media implemented in any method ortechnology for storage of information such as computer readableinstructions, data structures, program modules or other data. Computerstorage media includes, but is not limited to, RAM, ROM. EEPROM, sashmemory or other memory technology, CD-ROM, digital versatile disks (DVD)or other optical disk storage, magnetic cassettes, magnetic tape,magnetic disk storage or other magnetic storage devices, or any othermedium which can be used to store the desired information and which canbe accessed by computer 210. Communication media typically embodiescomputer readable instructions, data structures, program modules orother data in a transport mechanism and includes any informationdelivery media. The term “modulated data signal” means a signal that hasone or more of its characteristics set or changed in such a manner as toencode information in the signal. By way of example, and not limitation,communication media includes wired media such as a wired network ordirect-wired connection, and wireless media such as acoustic, RF,infrared and other wireless media. Combinations of any of the aboveshould also be included within the scope of computer readable media.

The system memory 230 includes computer storage media in the form ofvolatile and/or nonvolatile memory such as read only memory (ROM) 231and random-access memory (RAM) 232. A basic input/output system 233(BIOS), containing the basic routines that help to transfer informationbetween elements within computer 210, such as during startup, istypically stored in ROM 231. RAM 232 typically contains data and/orprogram modules that are immediately accessible to and/or presentlybeing operated on by processing unit 220. By way of example, and notlimitation, FIG. 2 illustrates operating system 234, applicationprograms 235, other program modules 236, and program data 237.

The computer 210 may also include other removable/nonremovablevolatile/nonvolatile computer storage media. By way of example only,FIG. 2 illustrates a hard disk drive 241 that reads from or writes tonon-removable, nonvolatile magnetic media, and an optical disk drive 255that reads from or writes to a removable, nonvolatile optical disk 256such as a DVD or other optical media. Other removable/non-removable,volatile/nonvolatile computer storage media that can be used in theexemplary operating environment include, but are not limited to,magnetic tape cassettes, sash memory cards, DVDs, digital video tape,solid state RAM, solid state ROM, and the like. The hard disk drive 241is typically connected to the system bus 221 through a nonremovablememory interface such as interface 240, and optical disk drive 255 aretypically connected to the system bus 221 by a removable memoryinterface, such as interface 250.

Alternatively, or in addition, the functionality described herein can beperformed, at least in part, by one or more hardware logic components.For example, and without limitation, illustrative types of hardwarelogic components that can be used include Field-programmable Gate Arrays(FPGAs), Program-specific Integrated Circuits (ASICs), Program-specificStandard Products (ASSPs), System-on-a-chip systems (SOCs), ComplexProgrammable Logic Devices (CPLDs), etc.

The drives and their associated computer storage media discussed aboveand illustrated in FIG. 2 , provide storage of computer readableinstructions, data structures, program modules and other data for thecomputer 210. In FIG. 2 , for example, hard disk drive 241 isillustrated as storing operating system 244, application programs 245,other program modules 246, and program data 247. Note that thesecomponents can either be the same as or different from operating system234, application programs 235, other program modules 236, and programdata 237. Operating system 244, application programs 245, other programmodules 246, and program data 247 are given different numbers here toillustrate that, at a minimum, they are different copies.

A user may enter commands and information into the computer 210 throughinput devices such as a keyboard 262, a microphone 263, and a pointingdevice 261, such as a mouse, trackball or touch pad. Other input devices(not shown) may include a joystick, game pad, satellite dish, scanner,or the like. These and other input devices are often connected to theprocessing unit 220 through a user input interface 260 that is coupledto the system bus, but may be connected by other interface and busstructures, such as a parallel port, game port or a universal serial bus(USB). A visual display 291 or other type of display device is alsoconnected to the system bus 221 via an interface, such as a videointerface 290. Video interface 290 can comprise a graphics card having aGPU. The GPU be used for computations. In addition to the monitor,computers may also include other peripheral output devices such asspeakers 297 and printer 296, which may be connected through an outputperipheral interface 295.

The computer 210 is operated in a networked environment using logicalconnections to one or more remote computers, such as a remote computer280. The remote computer 280 may be a personal computer, a hand-helddevice, a server, a router, a network PC, a peer device or other commonnetwork node, and typically includes many or all of the elementsdescribed above relative to the computer 210. The logical connectionsdepicted in FIG. 2 include a local area network (LAN) 271 and a widearea network (WAN) 273, but may also include other networks. Suchnetworking environments are commonplace in offices, enterprise-widecomputer networks, intranets and the Internet.

When used in a LAN networking environment, the computer 210 is connectedto the LAN 271 through a network interface or adapter 270. When used ina WAN networking environment, the computer 210 typically includes amodem 272 or other means for establishing communications over the WAN273, such as the Internet. The modem 272, which may be internal orexternal, may be connected to the system bus 221 via the user inputinterface 260, or other appropriate mechanism. In a networkedenvironment, program modules depicted relative to the computer 210, orportions thereof, may be stored in the remote memory storage device. Byway of example, and not limitation, FIG. 2 illustrates remoteapplication programs 285 as residing on remote computer 280. It will beappreciated that the network connections shown are exemplary and othermeans of establishing a communications link between the computers may beused.

Examples and Comparisons

To further illustrate certain embodiments of the disclosed methods,systems, and computer-readable storage media, and to provide variouscomparative analyses and data, below are some examples with comparisontest data.

Photos of a basic system, including a source, object grid, and detectorare shown in FIG. 3 . The x-ray source 201 used was a Comet MXR-160HP/11x-ray tube with a maximum energy of 160 kV. The maximum power was 1800 Wwith a 1 mm spot size or 800 W with a 0.4 mm spot size. In someembodiments, when the spot size is too large to resolve the object gridon the detector, a source grid is necessary. The relationship describingthe spot size above which the ability to resolve the object grating isreduced can be expressed as w_(source)≥(L₁+L₂)*P_(obj)/L₂, where w isthe spot size of the source, L₁ is the source-to-object distance, and L₂is the object-to-detector distance. Test data shown below used 1 mm Aland 0.1 mm Cu to filter the spectrum of the beam and reduce the flux atvery low photon energies. The working distance from source to detector203 was set at 2 m, and the detector used initially was a CMOS X-raydetector (e.g., Shad-o-box® 4k) with a 10 cm×10 cm field of view, 48 μmpixel pitch, and a Gd₂O₂S:Tb scintillator (Teledyne DALSA®). The objectand object grid 202 were placed halfway between the source and detector,resulting in a two-times magnification and good sensitivity to the smallangular deflections by the sample. The beam pattern was created withcommercial anti-scatter grids 204 used for medical imaging, consistingof parallel line patterns between 85 and 285 lines per inch (LPI).Typically, the grids are made of lead and aluminum sheets interspersed,but the finest grid was made with carbon fiber between lead sheets. Asecond grid, approximately half the object grid's spatial frequency, wasoriented such that the grid lines were perpendicular to those of theobject grid and was placed directly in front of the detector in order tocorrect for beam hardening, as described elsewhere herein.

Custom software was developed to handle both data acquisition andextraction of absorption, phase, and scatter images. Frames wereacquired from the detector, summed, and saved as floating-point tifffiles; this file type allowed viewing as an image while preserving rawnumbers from the detector and the full bit-depth of the detector. Theinterface to the data acquisition and processing software is shown inFIG. 4 .

Raw images 401 resembled a normal attenuation image, but with finevertical and horizontal lines visible from the object and detectorgrids. A Fourier-based processing method was used. First, a Fouriertransform 402 is taken of the image. For a system with parallel linegrids, peaks will be visible in the Fourier transform corresponding tothe spatial frequency of the grid. In FIG. 5 , peaks to the right andleft of center correspond to the first harmonic associated with theobject grid (which was oriented with its grid lines vertical), and thepeaks at top and bottom are the first harmonic of the detector grid(which was oriented with its lines horizontal. The region around thefirst harmonic when an object is present is compared with the sameregion a grid-only image (no sample). Pattern displacements of theobject grid were determined based on the phase values in a region of theFourier transform around the first harmonic. These were interpreted asrefraction and formed the differential phase contrast image. Patternvisibility, V, is calculated based on the ratio of the inverse Fouriertransform of the region around the first harmonic to the inverse Fouriertransform of the region around the zeroth harmonic, but can be moresimply understood as the amplitude of the projected grid pattern dividedby the mean. Pattern visibility reduction, V/V₀, which compares patternvisibility with an object present to pattern visibility without, wasinterpreted as scatter. Note, visibility can be reduced not only byscattering effects but also by beam hardening, since a harder spectrummay result in less modulation of the object grid. To correct for this, adetector grid was chosen that has similar attenuation properties as theobject grid so that it would be similarly affected by beam hardening.This grid was placed as close as possible to the detector to minimizethe effects of scatter on the projected pattern. The ratio of thescatter image to the apparent scatter in the detector grid imageproduced a correction to remove the portion of the visibility loss dueto beam hardening.

The first study performed was a test of the system sensitivity to theobject grid spatial frequency. Calibration standards were constructed ofiron oxide (i.e., Fe₃O₄) nanoparticles dispersed in epoxy at a 20%volume fraction. Objects were 6 mm thick and were constructed with twodifferent sizes of particles: 30 nm and 1 μm. A source spectrum with apeak energy of 40 kV was used. Results are shown in FIG. 6 , which showsthe corrected scatter ratio, which is the visibility reduction at theobject grid after correcting for beam hardening as a function of gridfrequency (in lines per inch). For the 1 mm particle sample, as the gridfrequency was increased, the scatter increased approximately linearly.For the 30 nm particles, the response was independent of grid frequency.This is consistent with theoretical descriptions of the measurementprocess where the x-ray interactions are due to Small Angle X-rayScattering (SAXS) and the response of the system is characterized by aparameter known as the correlation length. The correlation length of themeasurement at 40 kV ranges between 50 and 300 sm depending on the gridfrequency. For particles much larger than the correlation length,scatter signal increases with larger grating spatial frequencies, whilefor samples much smaller than the correlation length, scatter signalshould be independent of gratings frequency.

For explosives detection, particles larger than the correlation lengthare of primary interest. In certain embodiments, texture is consideredto be in the 1-1000 microns for what is defined as a powder. Thecorrelation length is inversely proportional to energy, so increasingthe spatial frequency of the object grid as much as possible willimprove measurement sensitivity. For signatures of explosive and benignmaterials that, when textured, typically have variations on lengthscales ranging from microns to millimeters in size, a finer object grid(higher spatial frequency) is generally advantageous. However, as gridfrequency decreases, imaging the projected grid pattern becomes moredifficult. This can be caused by the finite size of the x-ray sourceregion in the tube, finite resolution at the detector, and by limitedattenuation in a finely patterned grid (as spatial frequencies increase.Therefore, higher aspect ratio fabrication is required in order toretain sufficient thickness to modulate the beam). To examine theseeffects, we measured the visibility with no object present for severalgrids and measurement geometries. Higher visibilities indicate a largerfraction of the beam intensity is available for detecting refraction andscatter; lower visibilities will lead to noisier measurements. FIG. 7 isa graph that shows visibility as a function of endpoint energy for 103,120, and 230 LPI grids placed halfway between source and detector, and a120 LPI grid placed near the detector. Overall visibility values rangefrom less than 0.05 up to 0.30. Visibility decreases with energy, as thebeam becomes more penetrating, but changes relatively slowly at higherenergies. The coarsest grid, at 103 LPI, showed the highest visibilitiesand the finest grid, 230 LPI, showed the lowest. For the 120 LPI grid,measurements were done with the grid halfway between source anddetector, where pattern visibility was equally sensitive to resolutionlimitations due to the source and to the detector, and with the gridnear the detector, where resolution was strongly dependent on thedetector and independent of the source spot size. The grid showed highervisibility at the detector position, indicating that for grids halfwaybetween source and detector, the source spot size was the dominantfactor in reducing visibility.

The beam hardening correction was tested as a function of energy using a1.26 cm thick section of aluminum (for reference, the mean free path forattenuation in Al ranges from 0.68 cm at 40 keV to 2.7 cm at 160 keV).This was selected as a material that was expected to be homogeneous overany texture length scales that the measurement would be sensitive to(e.g., nm to μm). Fringe visibility reduction (V/V₀, where V₀ is thegrid visibility without an object present) is plotted in FIG. 8 as afunction of endpoint energy, both for the object grid alone (Aluncorrected), and for the object grid after beam-hardening correction bythe detector grid (Al corrected). We can verify that, although the rawobject grid (uncorrected) shows substantial decreases in fringevisibility, particularly at low energies, the correction by the detectorgrid results in calculated visibility reduction numbers independent ofpeak energy and consistent with a ratio of 1, indicating a homogeneous,non-scattering material, as expected.

The scatter for a typical gratings-based setup is expected to bedominated by small angle x-ray scattering (SAXS), a mechanism forelastic scattering that produces a spectrum as a function of ascattering vector reflecting the distribution of spatial features in thesample. For the gratings-based measurement, results are not resolved asa function of scattering vector, but sensitivity is greatest near thescattering angle defined by the object grid-to-detector distance, d, andthe size of the projected grid period. P_(projected). The momentumtransfer associated with this scattering angle can be related to acorrelation length in the material, ξ_(corr)=d*hc(P_(projected)·E),where E is the photon energy and E/hc is the photon wavelength; thiscorrelation length is closely related to the particle size that producespeak scatter intensity. In FIG. 9 , we plot corrected scatter ratio as afunction of particle size for a 60 kV spectrum. Solid lines aretheoretical estimates for a range of grating frequencies. As gratingfrequency was increased, the effects on scatter ratio become larger, andthe correlation length increased as well, leading to peak scattering atincreasingly large particle size. Measured results are shown for ironoxide nanoparticles at 7 nm, 30 nm, and 1000 nm mean particle size; theywere generally consistent with theoretical predictions.

The measurements shown established that fringe visibility is possible atenergies up to 160 kV and illustrated the importance of correcting forbeam hardening in interpreting fringe visibility reduction as scatter. Atradeoff is demonstrated between increased scatter signal for higherspatial frequency grids and reduced overall visibility as grid frequencyis increased, which will lead to lower signal-to-noise.

A source grid can improve the ability to resolve the object grating. Thesource grid is aligned parallel to the object grid, with half thespatial frequency (when the object grid was placed substantiallyequidistant between the source and detector); this results in multipleprojected images of the object grid that overlay at the detector. For a160 kVp spectrum and a 1 mm source spot size, adding a source grid with50% duty cycle and a period of 4.7 lines per mm increased the visibilityof a 9 lines per mm object grid placed 1 m downstream of the source and1 m upstream of the detector by approximately 4×. This approach allowsthe use of a larger spot size and therefore higher flux.

Referring to FIG. 10 , a composite image of a small bag (8″×9″), takenat 160 kV using a large spot size and a source grid to improve objectgrid visibility is shown according to embodiments described herein. Theimage was acquired using a system comprising a polychromatic source(e.g., Comet x-ray tube) run with a 1 mm spot size, a 103 LPI sourcegrid, 210 LPI object grid, and 103 LPI detector grid orientedperpendicular to the other two for beam hardening corrections, used withthe CMOS detector. The image on the left is an attenuation image. Theimage on the right is an attenuation image with scatter overlay (color).Propellants present in the bag show scatter, as do business cards, awatch band, and tic tac candies. Chapstick next to the model rocketengine looks similar by shape, but does not exhibit scatter.

In order to characterize the performance and verify consistency of animaging system, particularly the ability to detect texture, a set ofcalibration standards was developed. The first type of calibrationstandard comprises a scatter test object and provides a stable andrepeatable means for measuring scatter signal across different systems.The inventors determined that the scatter test object, which was stableand robust with well characterized small-scale structure, was beneficialso that the efficacy of different x-ray systems could be tested. Thescatter test object can have sufficient contrast for use at highenergies, when the cross section for elastic scatter is relativelysmall. One embodiment of the scatter test object calibration standardcomprises a block of polymer with microparticles or nanoparticlesdispersed evenly within it. The particles can comprise metal and/ormetal oxide. The particles have a known size distribution. The polymerblock can be fabricated with a series of steps or geometric features ofdifferent thickness. The resulting object provides a measure of x-rayscattering as a function of thickness, over a wide range of imagingsystems and x-ray energies, and which is stable and robust. The testingand calibration can be particularly advantageous for certainapplications including explosives detection and medical imaging. In suchapplications, the scatter test object can comprise a scatter-imagingphantom. The phantom is an object having the same scatter qualitiesand/or properties as a material in which one is interested in imaging.

In some embodiments, metal or metal oxide microparticles ornanoparticles were fixed in a polymer. Particles with a well-definedsize distribution are commercially available. This is important becausethe scatter signal exhibits sensitivity to the size of the particles ortexture. A polymer (e.g., epoxy) is robust and stable over time and addsno additional scatter signal. The metal or metal oxides have asufficiently high density that the x-ray refractive index change betweenthe particles and the polymer produce a strong scattering signal. Aftertesting numerous metal and metal oxide nanoparticles and microparticles,ZnO was found to disperse evenly in epoxy, and scatter step-wedges werecreated out of 20 vol % ZnO particles fixed in epoxy. Blocks werecreated with 1 μm particles, and with a distribution of particles 5 μmand below; steps were cut to be approximately 6 mm thick with a maximumthickness of 25 mm. The scatter step wedges are shown in FIG. 11A, alongwith scatter images (see FIG. 11B) at two different energies. Whentransitioning between different gratings-based systems, or differentenergies, contrast-to-noise was measured on the step wedges andcompared. The scatter test object calibration standard is not limited toa step wedge shape. A wedge with graded thickness is an example of analternative shape. Further still, the scatter test object can compriseone or more shapes having a constant thickness. A plurality of constantthickness calibration standards, each with different thickness,different particle loadings, and/or particle sizes can be used as analternative.

Another type of calibration standard comprises a beam hardening testobject for phase contrast x-ray imaging, which can be used to test forbeam hardening artifacts that can adversely affect the scattermeasurement and ensure the artifacts have been properly removed. The useof the beam hardening test object calibration standard relies on thefact that it does not have density fluctuations at length scales towhich the measurement is sensitive—that the materials in the calibrationstandard are homogeneous. This provides a baseline expectation that datataken with the test device will, if properly corrected for beamhardening, indicate no additional fringe visibility loss due to texture.For many applications, a beam hardening correction will be applied formultiple materials, spanning much of the periodic table, and formaterials with a wide range of attenuation values. The calibrationstandard is designed to contain multiple homogeneous materials across arange of atomic number, with the thickness of each material selected sothat a moderate amount of attenuation (10% to 90% of the original beam)is present.

One embodiment of the beam hardening test object calibration standardcan comprise three or more materials that are each homogeneous, with nolarge density variations on length scales between 10 nm and 200 microns,and represent a range of atomic numbers. The materials are machined to athickness suited to the energy of the x-rays used, such that 10-90% ofthe beam intensity is transmitted through the object. A corrected phasecontrast measurement, as described elsewhere herein, is performed withcorrections for spurious signals due to spectral changes duringattenuation, and the resulting scatter image of the calibration standardwill be consistent with background if the correction is successful.

In some embodiments, beam hardening test object comprises approximatelyone mean free path at 160 kV of aluminum (28 mm), stainless steel (7mm), copper (5.5 mm), and tin (1.0 mm). This gave a range of Z, andsubstantial attenuation, over which to test the beam hardeningcorrection. The beam hardening correction can correct partially thevisibility reduction observed in the calibration standard materials, incontrast with the complete correction observed with the 12.5 mm Alsample. This appears to be related to the relatively high attenuation ofthe calibration standard. Known homogeneous material samples (such aswater) show a corrected scatter value consistent with homogeneity.Accordingly, there is no issue with the calibration standardsignificantly impacting the measurements of the explosive and benignmaterials.

Measurements of a variety of materials, including threat and non-threatmaterials, were conducted in collaboration with Chuck Divin, Sabrina DePiero, Larry McMichael, and Harry Martz at Lawrence Livermore NationalLaboratory. These measurements were performed using a microfocus x-raytube (Hamamatsu L12161-07). The nominal spot size at max current was 50μm. The final measurement configuration is shown in FIG. 12 andconsisted of the Hamamatsu microfocus tube 1201 operated at 0.5 mAcurrent and 50 μm spot size, an object grid 1202 with 285 LPI (JPIHealthcare), and a detector grid 1203 with 120 LPI (Kiran Medical).Measurement time was necessarily very long—10 minutes, or 300 mA·s. Thiswas selected and verified to be well above the range where noise in theimages is dominated by counting statistics, in order to emphasize thephysical signatures. The small spot size provided by the microfocus tubenecessitated longer measurement times. In some embodiments, high-energy,large spot size measurements described herein have measurement timesthat are less than 10 minutes, 8 minutes, 6 minutes, 5 minutes, 3minutes, 2 minutes, 1 minute, or 30 seconds.

Data was acquired for two different spectra, chosen to be similar tospectra used for dual-energy measurements in current checkpointscreening. Calculated spectra are shown in FIG. 13 . The high energyspectrum, in blue, had an endpoint energy of 160 kV (but was reduced to150 kV with the Hamamatsu source), 2 mm of copper and 2 mm of aluminumfiltration. The average correlation length for this system, with the 285LPI grid, was 90 nm. The low energy spectrum, shown in red, had anendpoint energy of 100 kV, 2 mm aluminum filtration, and an averagecorrelation length of 170 nm.

The test dataset consisted of over 20 different benign materials,selected with knowledge of items typically found in baggage, and with awide range of densities, effective atomic numbers, and including severalitems which were powdery or had other density variations. Four threatmaterials were selected, all with some level of mesoscale texture. Threeof the materials were powders, with a range of grain sizes andpreparation methods, and one was a moldable. Materials were placed inplastic cylinders 3 cm in diameter and 2-3 cm thick, had a total mass of15-30 g per sample, and were imaged end-on to produce a large area withuniform thickness. An example vial 1402 and set of samples 1401assembled for imaging are shown in FIG. 14 .

TABLE 1 Material μ_(H) (cm⁻¹) μ_(L) (cm⁻¹) μ_(L)/μ_(H) v_(H) (cm⁻¹)v_(L) (cm⁻¹) raspberry jelly 0.208 ± 0.008 0.285 ± 0.010 1.37 −2.00E−03± 1.38E−02 −2.25E−03 ± 2.49E−03 strawberry jelly 0.211 ± 0.006 0.287 ±0.007 1.36 −4.44E−05 ± 1.33E−02 −9.30E−04 ± 2.45E−03 coconut oil 0.170 ±0.004 0.285 ± 0.005 4.67   9.20E−05 ± 1.19E−02 −1.32E−03 ± 2.64E−03rubber cement 0.119 ± 0.001 0.150 ± 0.001 1.27 −1.86E−04 ± 1.13E−02  2.83E−04 ± 2.02E−03 Vaseline 0.130 ± 0.008 0.177 ± 0.013 1.36  5.21E−04 ± 5.81E−03   2.97E−04 ± 2.53E−03 Colgate 0.191 ± 0.005 0.320± 0.008 1.67   3.63E−03 ± 6.45E−03   1.46E−02 ± 3.08E−03 Olay sunscreen0.165 ± 0.006 0.308 ± 0.010 1.87   9.21E−03 ± 6.44E−03   3.33E−02 ±3.16E−03 pure honey 0.194 ± 0.003 0.297 ± 0.004 1.54 −5.31E−04 ±6.88E−03 −9.83E−04 ± 2.60E−03 apricot scrub 0.150 ± 0.005 0.237 ± 0.0081.58   2.69E−04 ± 7.31E−03   4.84E−03 ± 3.37E−03 kids sunscreen 0.211 ±0.005 0.520 ± 0.011 2.47   5.66E−02 ± 1.01E−02   1.70E−01 ± 9.20E−03 OldSpice 0.195 ± 0.004 0.425 ± 0.008 2.18   5.10E−03 ± 7.32E−03   1.62E−02± 4.40E−03 antiperspirant Mitchum deodorant 0.187 ± 0.005 0.339 ± 0.0081.82   2.47E−03 ± 6.72E−03   7.62E−03 ± 2.98E−03 Banana sunscreen 0.151± 0.005 0.229 ± 0.008 1.52 −1.60E−03 ± 6.28E−03 −7.49E−03 ± 2.30E−03Banana kids 0.185 ± 0.005 0.399 ± 0.006 2.15   1.95E−02 ± 6.82E−03  6.47E−02 ± 4.34E−03 sunscreen AW toothpaste 0.220 ± 0.006 0.366 ±0.008 1.67   4.27E−03 ± 7.17E−03   3.12E−02 ± 4.77E−03 sunflower oil0.133 ± 0.003 0.185 ± 0.003 1.39   2.44E−04 ± 5.76E−03   5.11E−04 ±3.06E−03 Nutella 0.139 ± 0.013 0.269 ± 0.025 1.94   3.55E−03 ± 5.90E−03  4.01E−02 ± 4.92E−03 water 0.139 ± 0.002 0.213 ± 0.004 1.53   6.36E−05± 5.77E−03 −1.08E−03 ± 2.08E−03 flour 0.121 ± 0.001 0.176 ± 0.002 1.46  6.74E−03 ± 6.55E−03   7.03E−02 ± 7.07E−03 powdered sugar 0.093 ± 0.0030.135 ± 0.004 1.45   1.44E−02 ± 5.84E−03   1.28E−01 ± 5.23E−03 MaterialA_6 0.184 ± 0.018 0.289 ± 0.005 1.57   2.59E−03 ± 6.57E−03   3.05E−02 ±4.01E−03 Material A_7 0.186 ± 0.003 0.280 ± 0.004 1.51   4.02E−03 ±5.64E−03   2.92E−02 ± 3.79E−03 Material A_5 0.193 ± 0.003 0.293 ± 0.0051.52   3.84E−03 ± 6.18E−03   3.14E−02 ± 3.98E−03 Material B_9 0.131 ±0.010 0.191 ± 0.007 1.46   1.05E−03 ± 7.08E−03   2.44E−02 ± 7.07E−03Material B_121-1A 0.135 ± 0.004 0.196 ± 0.006 1.45 −5.38E−04 ± 6.02E−03  2.94E−02 ± 7.36E−03 Material C_11 0.113 ± 0.002 0.169 ± 0.003 1.50  1.73E−02 ± 6.54E−03   1.15E−01 ± 7.03E−03 Material C_14 0.067 ± 0.0020.098 ± 0.004 1.46   1.62E−02 ± 4.80E−03   1.19E−01 ± 4.94E−03 MaterialC_13 0.066 ± 0.002 0.096 ± 0.005 1.45   1.30E−02 ± 5.03E−03   1.16E−01 ±5.52E−03 Material C_10 0.110 ± 0.002 0.165 ± 0.005 1.50   1.36E−02 ±8.81E−03   1.15E−01 ± 7.96E−03 Material C_12 0.113 ± 0.002 0.167 ± 0.0031.49   1.49E−02 ± 5.80E−03   1.14E−01± 7.25E−03 Material C_18 0.105 ±0.001 0.155 ± 0.003 1.48   3.19E−03 ± 8.05E−03   3.92E−02 ± 1.17E−02Material C_19A 0.119 ± 0.001 0.177 ± 0.001 1.49   1.77E−02 ± 1.03E−02  1.52E−01 ± 8.06E−03 Material C_7A 0.151 ± 0.011 0.219 ± 0.021 1.45  9.33E−04 ± 7.80E−03   3.66E−03 ± 7.30E−03 Material D_2 0.127 ± 0.0180.178 ± 0.004 1.41   1.35E−03 ± 6.99E−03   1.91E−02 ± 9.51E−03 MaterialD_1 0.120 ± 0.002 0.179 ± 0.004 1.49   1.89E−03 ± 6.70E−03   2.05E−02 ±1.04E−02 Material D_4 0.120 ± 0.003 0.180 ± 0.005 1.51   1.95E−03 ±6.53E−03   1.59E−02 ± 9.95E−03 Material D_3 0.121 ± 0.003 0.182 ± 0.0051.51   1.39E−03 ± 6.70E−03   1.90E−02 ± 1.00E−02

Once data were acquired, the absorption/refraction/scatter images wereextracted and the scatter image corrected for beam hardening. A regionof interest was chosen, typically including most of the material, andmean and standard deviation values were extracted for both attenuation(I/I₀) and scatter (V/V₀). The attenuation was converted into anattenuation coefficient μ=−ln(I/I₀)/t, where t represents measuredsample thickness, and μ is in units of mean free attenuation paths percm (cm⁻¹). In analogous fashion, we extracted a scatter coefficientν=−ln(V/V₀)/t, in units of mean free scatter paths per cm (cm⁻¹). Bothquantities are implicitly weighted averages over all energies present inthe spectrum. The variations observed in both absorption and scatterimages were propagated as errors to obtain variation estimates for bothquantities. This was performed for both the high and low energy spectra,and the quantitative results are shown in Table I: attenuationcoefficient μ at low and high energy, and scatter coefficient ν at lowand high energy, for each sample measured.

The attenuation coefficients extracted from the phase contrastmeasurements can be interpreted in the same manner as conventional dualenergy measurements, which are analyzed to estimate effective atomicnumber Z_(eff) and density ρ. Here, we examined the ratio of the lowenergy and high energy attenuation coefficients (μ_(L)/μ_(H)); for afully calibrated system this quantity can be related to the effectiveatomic number Z_(eff). In FIG. 15 we plot μ_(L)/μ_(H) as a function ofμ_(H), which can be mapped approximately to density. Error bars indicatethe variation observed within each sample.

Benign materials were shown in black. Water was indicated at μ_(H)=0.139and μ_(L)/μ_(H)=1.53; a number of other benign materials had Z_(eff)similar to water, but at higher densities many of the benign materialsalso exhibited higher Z_(eff). Threat materials were labeled by letters:with each letter signifying a single material, but samples within eachgroup may have different preparation conditions. Material A showed aZ_(eff) similar to water but the density is significantly larger.Materials B and D were close to water in density and Z_(eff), althoughslightly lower in both. Material C exhibited a wide range of densitiescorresponding to different preparation conditions, but Z_(eff) stillclose to that of water. In all four material categories, at least someof the samples exhibited density/Z_(eff) which were consistent withbenign materials.

Next, we examined material properties revealed by scatter, plotting thescatter coefficient for the lower energy spectrum, ν_(L), as a functionof density (approximated as pH), shown in FIG. 16 . Note that a scattercoefficient of zero indicates no scatter was detected from a givenmaterial. The first qualitative observation we can make is that thisplot is distinct from the dual energy information, with the distributionof properties clearly different than the previous plot, indicating thatunique information is being displayed. For Materials A, B, and D, asmall but significant amount of scattering is present. Note that in mostcases, this distinguishes them from materials with otherwise similardensity and atomic number. Material C covers not only a wide range ofdensities, but also a wide range of textures, from apparentlyhomogeneous to very highly scattering, depending on the preparationmethod used.

The range of scatter values in the benign materials can be helpful fordiscrimination as well. At low density, powdered sugar (μ_(H)=0.093,ν_(L)=0.13) exhibits very high scattering; flour is also fairly highlyscattering (μ_(H)=0.12, ν_(L)=0.07). Nutella® exhibits a moderate amountof scattering (μ_(H)=0.14, ν_(L)=0.04). There were four different typesof sunscreen, which illustrate an interesting range of scatteringproperties. One of the sunscreens, Banana Boat® (μ_(H)=0.15,ν_(L)=−0.007), is an organic sunscreen and contains no metals; it isrelatively low in density and no significant scatter is observed. Olay®sunscreen (μ_(H)=0.17, ν_(L)=0.033) contains 3% ZnO particles andexhibits some scatter. Banana Boat Kids® contains 6% TiO₂ and 4% ZnO andshows higher scatter yet (μ_(H)=0.19, ν_(L)=0.065). The final sunscreen(Badger® brand kids sunscreen) shows high Z_(eff), high density, andhigh scatter (μ_(H)=0.21, ν_(L)=0.17): it includes 19% ZnO particles,nearly as high of a concentration as our scatter step wedges. Othermaterials which show a small amount of scatter include deodorants andtoothpaste, as can be seen in Table I. Note that materials which arehomogeneous, such as water, sunflower oil, honey, and Vaseline®, displayscatter values consistent with zero, confirming that the beam hardeningcorrection process is successfully accounting for fringe visibilitychanges associated with spectral changes.

For the higher energy spectrum, the absolute values of all the scattercoefficients are reduced, as shown in FIG. 17 . Several of the moreweakly scattering samples cannot be distinguished from non-scatteringmaterials, but the more strongly scattering materials are stilldistinguishable. Detailed numerical results can be seen in Table I.

Often, not all pixels on the detector will record x-rays. Scratches orother damage can leave patterns of dead pixels. If not corrected, or ifcorrected using linear interpolation which is typically used for x-rayimaging, this can introduce artifacts into the reconstructed scatter andphase contrast images. The grating is typically aligned with thedetector array, producing a pattern of vertical stripes. The gratingperiod is typically chosen so that its period on the image is a fewpixels. The image may show vertical intensity oscillations with a longerperiod, which might be a Moire pattern between the grating shadows andthe pixel boundaries. The value of any given pixel (x, y) of the gratingimage will be designated g(x, y), and the image size will be N_(x)×N_(y)pixels. FIG. 18A includes an image of a region of the grating image,showing the vertical grating lines and the longer period Moire pattern.Near the center is a line of bad pixels from a scratch on the detector.FIG. 18B is a grating image with both the object and detector gratingsin a section of bad pixels.

FIGS. 19A-19C include Fourier transforms of grating images. In FIG. 19A,a section of the Fourier transform of the grating image shows thecentral peak at (0, 0) and the first order harmonic peaks on either sideat (±613, ±19). The non-zero y values of the first harmonic peakindicate that the grating was not perfectly aligned with the detector inthis image. In FIG. 19B, a close-up of a region of the center of theFourier Transform of the grating image shows the central peak at (0, 0)and Moire peaks at (±10, 0). In FIG. 19C, a close-up of the regionaround the first harmonic shows the harmonic peak at (613, −9) andconvolutional Moire peaks at (603, −9) and (623, −9).

The Fourier transform of the grating image shows peaks at the origin andnear the x-axis representing the Moire period, the grating period, andat intervals of the Moire period on either side of the grating period.If the grating period is long enough, higher harmonics of the gratingwill show up although typically the grating spacing and placement arechosen so that only the first harmonic appears. If a detector grid ispresent, peaks will be present at the detector grid period and itsharmonics, as well as cross-harmonics between the detector and objectgrid. The grating Fourier transform will be denoted G(x, y).

After subtracting the dark image, bad pixels typically have values nearzero. The grating image, despite its features, is usually relatively lowcontrast. A simple threshold cut-off on the grating image is usuallyeffective for selecting pixels to fix.

The bad pixel detection can be made even better by reducing majorsources of large period variation within the grating image. A copy G′(x,y) is made of the Fourier transform of the grating image, and theregions around the central peak and the grating harmonic peak pairs areremoved, making sure to include the Moire satellites around the majorpeaks. Let p_(M) be the Moire period in the grating image, (p_(ox),p_(oy)) be the location of the first harmonic peak of the object grid,and (p_(dx), p_(dy)) be the first harmonic peak of the detector grid. Anacceptable filter is to choose r₁=2 p_(M) and r₂=10 p_(M), and then use:

$\begin{matrix}{{r\left( {x,y} \right)} = \sqrt{x^{2} + y^{z}}} & (1)\end{matrix}$ $\begin{matrix}{{F\left( {x,y} \right)} = \left\{ \begin{matrix}1 & {{r\left( {x,y} \right)} > r_{2}} \\0 & {{r\left( {x,y} \right)} > r_{1}} \\{\frac{1}{2}\left\lbrack {1 - {\cos\left( {{\pi\left( {{r\left( {x,y} \right)} - r_{1}} \right)}/\left( {r_{2} - r_{1}} \right)} \right)}} \right\rbrack} & {otherwise}\end{matrix} \right.} & (2)\end{matrix}$ $\begin{matrix}{{G^{\prime}\left( {x,y} \right)} = {{G\left( {x,y} \right)}{\prod\limits_{j = {- n_{o}}}^{n_{o}}{\prod\limits_{k = {- n_{d}}}^{n_{d}}{F\left( {{x + {jp}_{ox} + {kp_{dz}}},{y + {jp_{oy}} + {kp_{dy}}}} \right)}}}}} & (3)\end{matrix}$where n_(o) is the number of harmonic peaks of the object grid and n_(d)is the number of harmonic peaks of the detector grid.

This is then Fourier transformed back to give the image g₀ (x, y). Thebad pixels, being isolated aperiodic features, are composed primarily ofhigh frequency components so the inverse Fourier transform preservesthese structures. Since the zero-period component has been removed, theaverage value of the image will be zero. With most structure removed,almost all pixels will have values near zero while the dead pixels willhave highly negative values. All pixels with values lower than athreshold value will be considered bad and removed. A reasonablethreshold is −G(0, 0)/3.

Because Fourier analysis based on convolutional patterns around theharmonic peaks are used for producing the scatter and phase contrastimages, simply replacing a bad pixel by the average of its neighbors isinsufficient. This neglects the short scale variation on the order ofthe grid pattern that is crucial to the analysis. Instead, we will usethe idea that in the vicinity of any pixel out to a radius of a fewpixels, the pixel values can be approximated by contributions from allharmonic and cross-harmonic peaks from the gratings. FIG. 20 includes aFourier transform of a grating image with both an object and detectorgrid, showing first and second harmonics as well as cross-harmonics.

$\begin{matrix}{{g\left( {x,y} \right)} \approx {\sum\limits_{j = {- n_{o}}}^{n_{o}}{\sum\limits_{k = {- n_{d}}}^{n_{d}}{b_{({jk})}{e^{2\pi{i({{{({{jp}_{ox} + {kp}_{dx}})}{x/N_{x}}} + {{({{jp}_{oy} + {kp}_{dy}})}{y/N_{y}}}})}}.}}}}} & (4)\end{matrix}$

The b_((jk)) are complex fitting parameters. Because g(x, y) is real,b₍₀₀₎ must be real and b_((−j-k))=b*_((jk)). However, there is no needto include these constraints in the algorithm, since unconstrainedlinear least squares fitting via singular value decomposition (SVD) isrobust, simple, and reliable. Since this fit extends to several pixelsaround the bad pixel, detailed features within this region can be washedout for the fit. However, the image analysis technique involves alow-pass filter so that these details will be lost anyway. Despite thematrix-like terminology, conceptually in the fitting process b is avector and the pair (jk) is treated as a single index. This can beaccomplished with a mapping of (jk) to an index 1 running from 0 to 1max=(1+2n_(o))(1+2n_(d)).

Choose a starting fitting step r_(f), as the number of pixels in x and yto either side of the bad pixel to include in the fit. This should bechosen so that 1+2r_(f) at least covers one grating period. In addition,if entire columns of bad pixels are expected, it should be at least 2 toavoid ill-conditioned fits. For any given bad pixel at (x_(b), y_(b)),scan over all pixels (x, y) such that x_(b)−r_(f)≤x≤x_(b)+r_(f) andy_(b)−r_(f)≤y≤y_(b)+r_(f). If the n^(th) scanned pixel (starting fromn=0) is not bad, add its value g(x_(n), y_(n)) to the vector h of nearbygood pixel valuesh _(i) =g(x _(n) ,y _(n))  (5)

and add a row to the matrix of fitting vectors AA _(i,l) =e ^(2πi((jp) ^(ox) ^(+kp) ^(dx) ^()x) ^(n) ^(/N) ^(x) ^(+(jp)^(oy) ^(+kp) ^(dy) ^()y) ^(n) ^(/N) ^(y)) .  (6)

Let h have m elements, so that A is m×l max in size. If m<l_(max), thefit will be ill-conditioned—there will be more fit variables b_(j) thanconstraints h_(i). Even m=l max is likely to lead to poor results. Areasonable criterion is to have the problem over-determined by a factorof 2. If m<6, increase r_(f) by 1 and find h and A again; repeat untilm≥2 l_(max).

The problem is now a complex linear fit, h=A·b. This is solved byfinding the SVD of A.A=U·W·Vwhere V is a l_(max)×l_(max) unitary matrix, W is a l_(max)×l_(max)non-negative real diagonal matrix, and U is a m×l_(max) matrix which iscolumn orthonormalu _(j) ·u _(k)=δ_(jk),u_(j) is the j^(th) column of U, and the symbol indicates the innerproduct. The elements on the diagonal of W are called the singularvalues. The condition number of A is the ratio of the largest to thesmallest of the singular values. If the condition number of W is morethan 10, increase r_(f) by 1 and find h and A again.

Let W⁻¹ be the pseudo-inverse of W; a l_(max)×l_(max) diagonal matrixsuch that

${\overset{\sim}{W}}_{ii}^{- 1} = \left\{ \begin{matrix}{{0{if}\ W_{ii}} = 0} \\{1/W_{ii}\ {otherwise}}\end{matrix} \right.$The fit vector is then found byb=V·{tilde over (W)} ⁻¹ ·U ^(†) ·h.(It is worth noting that since we demand the condition number is finite,all W_(ii) will be non-zero so that W⁻¹=W⁻¹.) Finally, set

${g\left( {x_{b},y_{b}} \right)} = {\sum\limits_{l = 0}^{l_{\max}}{b_{ie}2\pi{i\left( {{\left( {{jp}_{ox} + {kp_{dx}}} \right)x_{b}/N_{x}} + {\left( {{jp_{oy}} + {kp_{dy}}} \right)y_{b}/N_{\mathcal{y}}}} \right)}}}$

Then repeat the procedure where h is filled with pixel values from thegrating+object image, replacing the bad pixel in the grating+objectimage with the fitted value (A does not change between the grating andgrating+object image, so it can be re-used and its SVD does not have tobe re-computed).

If the grating absorption modulation is low compared to the averagevalue of the image, the cross-harmonic peaks can be neglected. Forimages with a detector grid, a fairly long grating period (resulting inmultiple harmonic peaks) in one or both grids, and many bad pixels, thiscan potentially result in significant time savings due to the O(ml_(max)²) scaling of SVD.

FIGS. 21 and 22 give examples of the bad pixel detection (FIGS. 21A and22A) and correction algorithm (FIGS. 21B and 22B). FIGS. 23 and 24 showhow fixing bad pixels improves the reconstructed phase and scatterimages (FIGS. 23A and 24A are before correction, FIGS. 23B and 24B areafter correction), respectively. FIG. 25 is an example of bad pixelcorrection with both an object and detector grating and withcontributions from multiple harmonics from both gratings (FIG. 25B isafter bad pixel correction).

In some embodiments, the signal-to-noise ratio in the acquired data canbe significantly improved by tilting the source grating, the objectgrating, the detector grating, or combinations thereof. Tilting caneffectively make line sources narrower without the burden of physicallymanufacturing gratings with extremely narrow grating elements (i.e.parallel channels with high aspect ratios). Tilting can compriserotating the gratings about an axis parallel to grating element lines.In certain embodiments wherein the grating elements are parallelchannels (each channel having a width and a height), the amount of tiltis greater than zero degrees and less than or equal to a maximum angleequivalent to the arctangent of the width of the channel divided by theheight.

Referring to FIG. 25 , a schematic illustration depicts a grating 2501having 210 parallel lines per inch and utilizing a 2 degree tiltrelative to an incident beam 2502 from an x-ray source. Experimentallyobserved visibility changes with grid tilt were simulated and thesimulation results are discussed herein. The geometry and materials ofthe measurement conditions were set up in a transport model to calculatethe total flux at the detector for various grid tilt (i.e., rotation)angles. The flux in an array of 48 μm pixels is tallied with MCNP 6.1—aMonte Carlo transport code to determine the pixel-based amplitude toaverage flux ratio. While the physics of the SAXS signal induced by atest object is not included in the purely atomic number-atomic mass(ZAID) based cross section libraries, the no-object visibility (toundergo reduction from object scatter) can be computed. Ideally thisquantity should be as large as is obtainable for the greatestsensitivity.

FIGS. 26A and 26B show modeled flux at the detector from a 0.4 mm spotsize 100 kVp source showing values of fringe visibility fits (26A), andplot of fringe visibility (H₁/H₀) along with the second harmonic ratio(H₂/H₀) as a function of rotation angle (27B). FIG. 26A shows themodeled flux at the detector for various rotation angles. The gridvisibility for each angle is determined from a fit to the first twoharmonics and is plotted in FIG. 26B. As is in the measurement, a localminimum in visibility appears at zero degrees and increasessymmetrically for angular changes from zero with a maximum value justbefore the fully transmitted grid openings go to zero.

This behavior for x-rays emitted from a 0.4 mm spot at 2 m from thedetector is very unexpected. A hint at its underlying origin is revealedby considering an idealized signal from a 0 mm spot (FIG. 27A), or evenmore strikingly from an idealized parallel beam (FIG. 27B). FIG. 28Bshows that an obvious consequence of grid rotation is an effectivereduction of transmitted duty-cycle. While a reduction of the fringeamplitude or first harmonic can be seen for increasing |θ| (smallereffective duty-cycle), the fringe average is reduced by an even greateramount resulting in increased ratio visibility and thus increasedscatter sensitivity. Additionally, in the limit of parallel beam andperfect attenuation of the lead, an analytical relationship for harmonicratios is simply

$\left( \frac{H_{n}}{H_{0}} \right)_{\parallel} = {\frac{P}{\pi ns}\sin\frac{\pi ns}{P}}$

where s is the effective slit width for the rotated grid. Assumingperfect attenuation for the lead absorber regions the effective slitwidth as a function of rotation angle is given by

${\frac{s}{P} = {\frac{1}{2} = {{- \frac{T}{2s_{1\text{:1}}}}{\theta }}}},$

where P is the grid period, T is the grid thickness, s 1:1 is the widthof the transmitting slits for the 1:1 duty-cycle grid and θ is the“small” rotation angle. Also, the effective duty-cycle (d_(c) :1) canthen be expressed with

$d_{c} = {\frac{s}{P - s}.}$In the parallel beam case it is interesting to note that the firstharmonic ratio is a maximum when both s/P and the intensity go to zero.

This behavior is relaxed by the more appropriate non-parallel beam andfinite spot size physics. The s/P equation above still assumes perfectattenuation, but not a parallel beam and can be used as an approximatemapping from simulated rotation to simulated duty-cycle. A consequenceof this mapping is that the analytical relationship for harmonic ratiosabove can be empirically modified to give a reasonably accurate fit tothe non-parallel beam data with

$\frac{H_{1}}{H_{0}} = {a_{0}\frac{\sin\left\lbrack {a_{3}\left( {1 - {a_{1}{\theta }} + {a_{2}\theta^{2}}} \right)} \right\rbrack}{1 - {a_{1}{\theta }} + {a_{2}\theta^{2}}}}$where a_(j) are four fit parameters. Interpreting a₁ as T/s_(1.1) andgiven θ for the maximum H₁/H₀, the equations for s/P and for d_(c), canbe used to infer the grid duty-cycle having maximum sensitivity. FIG. 28shows that the modified harmonic ratio gives very reasonable fits torotated object grid measurements indicating that the effect is due to achange in duty-cycle even when the source deviates from a parallel beam.Using the modified harmonic ratio fits, the maximum fringe visibilitygives an approximate effective duty-cycle 1:7 for both the 210 LPI gridsand 1:4 for the 230 LPI grid.

In view of the many possible embodiments to which the principles of thedisclosed invention may be applied, it should be recognized that theillustrated embodiments are only preferred examples of the invention andshould not be taken as limiting the scope of the invention. Rather, thescope of the invention is defined by the following claims. We thereforeclaim as our invention all that comes within the scope and spirit ofthese claims.

What is claimed is:
 1. A method comprising: emitting x-rays from anx-ray source; forming a patterned beam using an object grating placedproximal to an object to be imaged and between the x-ray source and adetector grating; acquiring through the detector grating a first imagewith the object and a second image without the object; measuringvisibilities of the object grating from the first and second images todetermine an object grating visibility reduction due to scatter and beamhardening; measuring visibilities of the detector grating from the firstand second images to determine a detector grating visibility reductiondue to beam hardening; and applying a beam hardening correction based ona comparison of the object grating visibility reduction and the detectorgrating visibility reduction to generate a corrected scatter image. 2.The method of claim 1, comprising creating a series of periodicallyrepeating apparent sources from the x-rays using a source gratingsituated proximal to the x-ray source, wherein the patterned beam isformed by patterning the series of periodically repeating apparentsources.
 3. The method of claim 2, wherein the source, object, anddetector gratings have respective grating elements, wherein the objectgrating is placed at distances L₁ from the source grating and L₂ fromthe detector grating, wherein the periodicities, P, of the source andobject grating elements are related byP_(source)=P_(object)*[(L₁+L₂)/L₂] and wherein the source and objectgrating elements are substantially parallel.
 4. The method of claim 3,wherein the detector grating elements are oriented substantiallyorthogonally relative to the object grating elements and a beam axis andwherein the object grating and the detector grating have a substantiallyequivalent x-ray attenuating factor.
 5. The method of claim 2, furthercomprising tilting the source grating by rotating the grating about anaxis parallel to grating element lines.
 6. The method of claim 2,wherein the acquiring includes detecting the images with a detector,wherein the object grating is approximately equidistant between thesource and the detector.
 7. The method of claim 6, wherein the detectorgrating has a periodicity, P_(detector), equivalent to that of thesource grating, P_(source).
 8. The method of claim 2, wherein the sourcegrating, object grating, detector grating, or combinations thereof havegrating elements comprising a parallel line pattern.
 9. The method ofclaim 1, wherein the emitting the x-rays comprises emitting the x-raysfrom a polychromatic source operating at an endpoint energy greater thanor equal to 100 keV and generating a spot size greater than or equal to0.5 mm.
 10. The method of claim 9, further comprising operating thepolychromatic source at an endpoint energy greater than or equal to 150keV, 160 keV, 175 keV, 200 keV, or 450 keV.
 11. The method of claim 1,further comprising tilting the object grating and detector grating byrotating the gratings about an axis parallel to grating element lines.12. The method of claim 1, wherein the object and detector gratingscomprise an equivalent material and have an equivalent thickness. 13.The method of claim 1, wherein the object to be imaged is a scatter testobject calibration standard and further comprising performing acalibration of x-ray scatter, the scatter test object calibrationstandard comprising metal or metal oxide particles distributed in apolymer matrix and having a stepped-wedge geometry of at least threedifferent thicknesses.
 14. The method of claim 1, wherein the object tobe imaged is a beam hardening test object calibration standard andfurther comprising performing a calibration of beam hardening, the beamhardening test object calibration standard comprising three or morehomogeneous materials in a range of atomic numbers, with no largedensity variations on length scales between 10 nm and 200 microns, andhave a thickness such that 10-90% of the x-ray intensity is transmittedthrough the test object.
 15. A system comprising: an x-ray sourceconfigured to provide source x-rays; a detector; an object grating and adetector grating, wherein the object grating is proximal to a positionof an object to be imaged and situated between the x-ray source and thedetector grating; and processing circuitry operably connected to thedetector and configured to execute computer-readable instructions to:acquire through the detector grating a first image with the object and asecond image without the object; measure visibilities of the objectgrating from the first and second images to determine an object gratingvisibility reduction due to scatter and beam hardening; measurevisibilities of the detector grating from the first and second images todetermine a detector grating visibility reduction due to beam hardening;and apply a beam hardening correction based on a comparison of theobject grating visibility reduction and the detector grating visibilityreduction to generate a corrected scatter image.
 16. The system of claim15, further comprising a source grating configured to create a series ofperiodically repeating apparent sources from the source x-rays, whereinthe object grating is configured to pattern the series of periodicallyrepeating apparent sources into a patterned beam.
 17. The system ofclaim 16, wherein the source, object, and detector gratings haverespective grating elements, wherein the object grating is placed atdistance L₁ from the source grating and distance L₂ from the detectorgrating, wherein the periodicities of the source and object gratingelements are related by P_(source)=P_(object)*[(L₁+L₂)/L₂].
 18. Thesystem of claim 17, wherein the detector grating elements are orientedorthogonally relative to the object grating elements and a beam axis andwherein the detector and object gratings having an equivalent x-rayattenuation factor.
 19. The system of claim 17, wherein the objectgrating and detector grating are positioned such that the object gratingelements and the detector grating elements are tilted by a rotation ofthe gratings about an axis parallel to grating element lines.
 20. Thesystem of claim 16, wherein the source grating is positioned such thatsource grating elements are tilted by a rotation of the grating about anaxis parallel to grating element lines.
 21. The system of claim 16,wherein the object grating is positioned approximately equidistantbetween the source and the detector.
 22. The system of claim 21, whereinthe detector grating has a periodicity, P_(detector), equivalent to thatof the source grating, P_(source).
 23. The system of claim 16, whereinthe source grating, object grating, detector grating, or combinationsthereof have grating elements comprising a parallel line pattern. 24.The system of claim 15, wherein the x-ray source comprises apolychromatic source configured to operate at an endpoint energy greaterthan or equal to 100 keV and a spot size greater than 0.5 mm.
 25. Thesystem of claim 24, wherein the polychromatic source is configured toprovide source x-rays at an endpoint energy greater than or equal to 150keV, 160 keV, 175 keV, 200 keV, or 450 keV.
 26. The system of claim 15,wherein the detector grating abuts the detector.
 27. The system of claim15, wherein the object and detector gratings comprise an equivalentmaterial and have an equivalent thickness.
 28. A non-transitory computerreadable storage medium storing one or more programs, the one or moreprograms comprising instructions, which when executed by one or moreprocessors operably connected to an x-ray imaging system that comprises:an x-ray source; an object grating proximal to a position of an objectto be imaged; and a detector grating having detector grating elementsthat are oriented orthogonally relative to object grating elements and abeam axis; cause the x-ray imaging system to: acquire through thedetector grating a first image with the object and a second imagewithout the object; measure visibilities of the object grating from thefirst and second images to determine an object grating visibilityreduction due to scatter and beam hardening; measure visibilities of thedetector grating from the first and second images to determine adetector grating visibility reduction due to beam hardening; and apply abeam hardening correction based on a comparison of the object gratingvisibility reduction and the detector grating visibility reduction togenerate a corrected scatter image.
 29. The non-transitory computerreadable storage medium of claim 28 storing one or more programs, theone or more programs comprising instructions, which when executed by oneor more processors operably connected to the x-ray imaging system causethe x-ray imaging system to perform a calibration, wherein the object tobe imaged is a scatter test object, a beam hardening test object, orboth.